Monday, December 10, 2007

Carl Jacobi: 203rd birthday

Carl Gustav Jacob Jacobi was born on December 10th, 1804. This ancient Jewish Prussian string theorist was one of the greatest mathematicians of all times and one of the most inspiring mathematicians of his times.

As a classical liberal, he was active in the 1848 revolution. Also, he suffered a breakdown from overwork at the age of 39 but he lived for 8 more years.

Jacobi was a true champion of special functions. Jacobi polynomials are those hypergeometric series that actually terminate. Jacobi showed that his elliptic functions are omnipresent as solutions to many equations in mathematical physics such as the pendulum, a symmetric top in a gravitational field, and a freely spinning body. Jacobi also liked to compute the motion of planets. Jacobi's integral gives a limited solution to the three-body problem. The Hamilton-Jacobi equation relates the time-derivative of the classical action with the Hamiltonian and is one of the key insights of modern abstract classical mechanics.

He also applied elliptic functions to number theory. For example, he proved a couple of "simpler" theorems due to Fermat. In a different paper, he introduced the Jacobi symbol, a generalized Legendre symbol involving prime factorizations of numbers. All mathematicians and physicists know him for the Jacobian, the determinant of a matrix of partial derivatives of new coordinates with respect to old coordinates, and for the Jacobi identity relating three double commutators of operators which is a defining formula for Lie groups. Some people remember his iterative Jacobi method to solve sets of equations in linear algebra. The Carathéodory-Jacobi-Lie theorem allows one to patch-wise define coordinates and momenta on each symplectic manifold.

String theorists know him as a colleague primarily for his Jacobi theta functions, appearing in partition sums and correlators of worldsheet fields at one-loop level (toroidal worldsheets). He has demonstrated the Jacobi triple product, the identity that allows you to re-express infinite sums as infinite products. A modern elegant proof of this formula uses a simplified model of the Dirac sea, probably pioneered by Richard Borcherds, and random partitions as defined by Andrei Okounkov, a fresh Fields medal winner and a string-theoretical mathematician.

In 1841, Jacobi reintroduced the modern symbol for the partial derivatives, originally invented by Legendre. It became standard.

But I chose the most string-theoretical result of Jacobi's life as the ultimate punch line. In 1829, he proved a "rather obscure formula" ("aequatio identica satis abstrusa") involving three terms that are fourth powers of a Jacobi theta function which verifies the spacetime supersymmetry at the level of partition sums in the Ramond-Neveu-Schwarz formulation of superstring theory. Mankind had to wait for nearly 150 years, until the late 1970s, to see what is the true physical reason why this fascinating identity holds and to make the formerly obscure formula transparent.